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A survey of numerical methods for stochastic differential equations

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Abstract

The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.

Suggested Citation

  • P. E. Kloeden & Eckhard Platen, 1989. "A survey of numerical methods for stochastic differential equations," Published Paper Series 1989-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    • Handle: RePEc:uts:ppaper:1989-1
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    File URL: https://link.springer.com/article/10.1007/BF01543857
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    Cited by:

    1. Tuckwell, Henry C. & Jost, Jürgen, 2012. "Analysis of inverse stochastic resonance and the long-term firing of Hodgkin–Huxley neurons with Gaussian white noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5311-5325.
    2. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous‐Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, June.
    3. Ogawa, Shigeyoshi, 1995. "Some problems in the simulation of nonlinear diffusion processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 217-223.
    4. Yoshihiro Saito & Taketomo Mitsui, 1993. "Simulation of stochastic differential equations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(3), pages 419-432, September.
    5. Kamal Boukhetala & Arsalane Guidoum, 2011. "Sim.DiffProc: A Package for Simulation of Diffusion Processes in R," Working Papers hal-00629841, HAL.
    6. R. Biscay & J. Jimenez & J. Riera & P. Valdes, 1996. "Local Linearization method for the numerical solution of stochastic differential equations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(4), pages 631-644, December.
    7. Hu, Rong, 2020. "Almost sure exponential stability of the Milstein-type schemes for stochastic delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).

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